Missing Date

The mean of thirteen-item data set is 320. Eleven of the items are 300, 320, 199, 175, 325, 520, 156, 225, 326, 421, and 504. The median is 325. Find the remaining two items in the set if it is known that these items have the greatest possible difference.

The mean of thirteen-item data set is 320. Eleven of the items are 300, 320, 199, 175, 325, 520, 156, 225, 326, 421, and 504. The median is 325. Find the remaining two items in the set if it is known that these items have the greatest possible difference.

let's arrange these numbers in increasing order, we get:

156, 175, 199, 225, 300, 320, 325, 326, 421, 504, 520

now, 325 is the median. since there should be 13 numbers, we need to have 6 numbers on either side of 325. we already have 6 on the left of 325, so with the addition of the two missing numbers we will have 6 on the right of 325. so a tentative list of our numbers is as follows:

now we are also told that the numbers x and y have the greatest difference, so, one has to be the smallest possible number either can be. what is the smallest number either can be? well, the median is 325, so to be on the right of 325 when arranged in ascending order, one of the numbers have to be at least 325, that is, greater than or equal to 325. so the least it can be is 325.